Theorem mpt2eq123dv 5567

Description: An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.)

Hypotheses

Ref	Expression
mpt2eq123dv.1	|- ( ph -> A = D )
mpt2eq123dv.2	|- ( ph -> B = E )
mpt2eq123dv.3	|- ( ph -> C = F )

Assertion

Ref	Expression
mpt2eq123dv	|- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)   D( x, y)   E( x, y)   F( x, y)

Proof of Theorem mpt2eq123dv

Step	Hyp	Ref	Expression
1		mpt2eq123dv.1	. 2 |- ( ph -> A = D )
2		mpt2eq123dv.2	. . 3 |- ( ph -> B = E )
3	2	adantr 261	. 2 |- ( ( ph /\ x e. A ) -> B = E )
4		mpt2eq123dv.3	. . 3 |- ( ph -> C = F )
5	4	adantr 261	. 2 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> C = F )
6	1, 3, 5	mpt2eq123dva 5566	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  mpt2eq123i  5568